# Is "$(\exists z)$ the sky is blue" a proposition?

Is "$$(\exists z)$$ the sky is blue" a proposition? I am unsure as this sentence certainly states something which could be either true or false; however the $$(\exists z)$$ seems meaningless here- I am unsure if the fact that $$z$$ is unused after it has been quantified is relevant here.

Thanks

It is a proposition. Here the predicate $$P(z)$$ would be the statement "the sky is blue." Although this has nothing to do with $$z$$, it is still a valid predicate in $$z$$ (analogous to a constant map from one set to another).

Does there exist some object $$z$$ such that the sky is blue? Why yes, no matter what object $$z$$ you pick, the sky is still blue. So in fact, we have the more general proposition: "$$(\forall z)\text{ the sky is blue.}$$"

Of course, this is assuming the sky really is always blue...

• A tautology is a propositional form that is always valid, whereas a validity is just that. There’s nothing tautological about “the sky is blue” even if it is always true. It is a validity if we take it as an axiom, but it’s not an entailment of FOL in general. May 30 at 4:23
• The proposition "for some object, the sky is blue" is not a tautology, which is a very strong claim, nor even a validity, merely true in the standard interpretation (our world). May 30 at 4:46
• "the predicate P(z) would be the statement "the sky is blue" "... There is no $z$ in the statement "the sky is blue". It is simply a statement $P$. The OP's question is quite confusing, mixing natural language and predicate-logic: in natural language it is quite unnatural to say "there is something such that the sky is blue", while in predicate logic there is nothing wrong in writing $\exists x \varphi$ also in case that $x$ is not free in $\varphi$. May 30 at 6:55

You’re referring to vacuous quantification. While it is kind of odd, vacuous quantification is innocuous/redundant at the level of syntax, and it simplifies things at the level of semantics.

That is, $$P$$, $$\exists x P$$, and $$\forall x P$$ are all equivalent for a formula $$P$$ with $$x$$ not free. Vacuous quantification then allows us to more easily define what constitutes a well-formed formula (wff); namely, if $$P$$ is a formula and $$x$$ a variable, then $$\forall x P$$ and $$\exists x P$$ are formulas.